Remark 3.1 
Observe that, using (REF ) and [1]}, the coefficients \({a}_n(s)\)  and \({b}_n(s)\)  can be written in the following way
\( {a}_n(s) = s\left(\int _0^{\infty }\frac{{\hat{P}}_n(s;y){W}(s;y){\hat{P}}_n^*(s;y)}{y}\mathrm {d}y\right){\gamma }_n(s),\\ {b}_n(s) = s\left(\int _0^{\infty }\frac{{\hat{P}}_n(s;y){W}(s;y){\hat{P}}_{n-1}^*(s;y)}{y}\mathrm {d}y\right){\gamma }_{n-1}(s).\) 
